Exercise 4: Clustering and classification
date()## [1] "Thu Nov 19 18:33:53 2020"library(tidyr); library(dplyr); library(ggplot2); library(gmodels)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionData description
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
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## select# load the data
data("Boston")
# explore the dataset
dim(Boston)## [1] 506 14str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...The Boston data set has 506 observations and 14 variables. The data include information on housing values in suburbs of Boston. Full list of variables and descriptions of variables are available here
Overview of the data
pairs(Boston)
pairs(Boston[c("crim","age","dis","rad","tax","black","medv")])
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00From the graphs we can see the distributions of variables and their relationship with other variables. When we take a closer look some selected variables, we see that for example crime rates are higher in areas where there are larger proportions of older houses or in areas where the mean of distances to employment centers is shorter.
Standartizatoin of data, creating test and train data sets
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# change the object to data frame
boston_scaled<-as.data.frame(boston_scaled)
# create a quantile vector of crim
bins <- quantile(boston_scaled$crim)
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label=c("low","med_low","med_high","high"))
# look at the table of the new factor crime
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# remove the crime variable from the data set
boston_scaled <- dplyr::select(boston_scaled, -crim)
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]After standardization, we can see from the summary table that all variables now have the mean zero and the minimum and maximum values of all scaled variables varies in much smaller intervals than in the original data.
After creating a categorical variable for crime rates, we can see from the table that observations are quite equally distributed in four categories.
Linear discriminant analysis
# linear discriminant analysis
lda.fit <- lda(crime~., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
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## Prior probabilities of groups:
## low med_low med_high high
## 0.2500000 0.2549505 0.2574257 0.2376238
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## Group means:
## zn indus chas nox rm age
## low 1.1119496 -0.9393112 -0.07742312 -0.9228920 0.50774701 -0.9268845
## med_low -0.0894819 -0.3220676 -0.11943197 -0.5637591 -0.12140935 -0.2542705
## med_high -0.3874684 0.1914499 0.18195173 0.3810147 0.05350906 0.4524743
## high -0.4872402 1.0172418 -0.02626030 1.0953787 -0.38049558 0.8305581
## dis rad tax ptratio black lstat
## low 0.9777487 -0.6828149 -0.7301005 -0.49716063 0.3779097 -0.79282024
## med_low 0.3418380 -0.5492448 -0.4968758 -0.04941921 0.3265930 -0.14629670
## med_high -0.3958355 -0.4374537 -0.3263755 -0.31434213 0.1258813 -0.00690559
## high -0.8766431 1.6368728 1.5131579 0.77931510 -0.7086547 0.91370918
## medv
## low 0.5729166
## med_low 0.0100506
## med_high 0.1624551
## high -0.6981924
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## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08872348 0.68193973 -0.87484315
## indus -0.01569326 -0.14566927 0.22168599
## chas -0.10333421 -0.03596710 -0.06392917
## nox 0.46760648 -0.62776859 -1.43362799
## rm -0.11347109 0.04220934 -0.19385778
## age 0.18319744 -0.40714029 -0.01801466
## dis -0.08108880 -0.12623037 0.10912809
## rad 3.30472334 0.99409237 0.11884968
## tax 0.09185883 -0.08054746 0.41822218
## ptratio 0.13104348 0.04212104 -0.21373412
## black -0.13775302 0.03263643 0.14442319
## lstat 0.28306455 -0.08897182 0.26415866
## medv 0.23577942 -0.31331235 -0.18260884
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## Proportion of trace:
## LD1 LD2 LD3
## 0.9489 0.0399 0.0112# target classes as numeric
classes <- as.numeric(train$crime)
#plot LDA biplot
plot(lda.fit, dimen = 2, col = classes, pch = classes)
From the output of our LDA model, we see that the first linear discriminant explain 94.5% of between group variance. This can be also seen from the plot where along the x axis (LD1) the separation of different groups is somewhat clearer, especially for the high category, than along the y axis (LD2).
Predictions
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 10 14 2 0
## med_low 4 14 5 0
## med_high 1 5 14 2
## high 0 0 0 31From the table we see that our LDA model did fairly good job as the majority of predictions were correct in most categories. The med low category was the most hardest to predict right.
K-means clustering
set.seed(123)
library(MASS); library(ggplot2)
data("Boston")
# center and standardize variables
boston_scaled <- scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)
# euclidean distance matrix
dist_eu <- dist(boston_scaled)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# k-means clustering:
km <-kmeans(boston_scaled, centers = 2)
pairs(boston_scaled, col = km$cluster)
pairs(boston_scaled[6:10], col = km$cluster)
# Investigate the optimal number of clusters with the total of within cluster sum of squares (tWCSS)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering: 3 clusters
km <-kmeans(boston_scaled, centers = 3)
pairs(boston_scaled, col = km$cluster)
pairs(boston_scaled[6:10], col = km$cluster)
When we first run K-means clustering with two clusters we can see for example rad and tax variables seem have a clear effect on the clustering results. When we investigate the optimal nubmer of clusters, three clusters seem to be enough according the total of within cluster sum of squares. With three clusters again rad and tax seem to affect the clustering results.
Bonus
library(MASS)
data("Boston")
set.seed(123)
# center and standardize variables
boston_scaled <- scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)
km <-kmeans(boston_scaled, centers = 3)
boston_scaled1<-data.frame(boston_scaled, km$cluster)
# linear discriminant analysis
lda.fit2 <- lda(km.cluster~., data = boston_scaled1)
# print the lda.fit object
lda.fit2## Call:
## lda(km.cluster ~ ., data = boston_scaled1)
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## Prior probabilities of groups:
## 1 2 3
## 0.2806324 0.3992095 0.3201581
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## Group means:
## crim zn indus chas nox rm
## 1 0.9693718 -0.4872402 1.074440092 -0.02279455 1.04197430 -0.4146077
## 2 -0.3549295 -0.4039269 0.009294842 0.11748284 0.01531993 -0.2547135
## 3 -0.4071299 0.9307491 -0.953383032 -0.12651054 -0.93243813 0.6810272
## age dis rad tax ptratio black
## 1 0.7666895 -0.8346743 1.5010821 1.4852884 0.73584205 -0.7605477
## 2 0.3096462 -0.2267757 -0.5759279 -0.4964651 -0.09219308 0.2473725
## 3 -1.0581385 1.0143978 -0.5976310 -0.6828704 -0.53004055 0.3582008
## lstat medv
## 1 0.85963373 -0.6874933
## 2 0.09168925 -0.1052456
## 3 -0.86783467 0.7338497
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim 0.03654114 0.20373943
## zn -0.08346821 0.34784463
## indus -0.32262409 -0.12105014
## chas -0.04761479 -0.13327215
## nox -0.13026254 0.15610984
## rm 0.13267423 0.44058946
## age -0.11936644 -0.84880847
## dis 0.23454618 0.58819732
## rad -1.96894437 0.57933028
## tax -1.10861600 0.53984421
## ptratio -0.13087741 -0.02004405
## black 0.15432491 -0.06106305
## lstat -0.14002173 0.14786473
## medv 0.02559139 0.37307811
##
## Proportion of trace:
## LD1 LD2
## 0.8999 0.1001# target classes as numeric
classes <- as.numeric(boston_scaled1$km.cluster)
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
#plot LDA biplot
plot(lda.fit2, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit2, myscale = 3)
From the plot we can see that rad and tax varaiables are mostly correlated with the first linear discriminant and dis and age with the second linear discriminant.
Super-bonus
set.seed(123)
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
#Next, install and access the plotly package. Create a 3D plot (Cool!) of the columns of the matrix product by typing the code below.
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
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## select## The following object is masked from 'package:ggplot2':
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## last_plot## The following object is masked from 'package:stats':
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## filter## The following object is masked from 'package:graphics':
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## layoutclasses <- as.numeric(train$crime)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=classes)## Warning: `arrange_()` is deprecated as of dplyr 0.7.0.
## Please use `arrange()` instead.
## See vignette('programming') for more help
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.I didn’t figure out anymore how to add clusters as colors.